Morph targets: A sphere shape turning into a cylindrical unwrapped sphere shape
I would do this backwards.

Crate a plane, enter edit mode (Tab) and subdivide it.

Press (U) and select Unwrap to create a UV map.



Exit edit mode Tab and in the properties panel create two shape keys by pressing the + box TWICE.That will create a base shape key for the plane and one for the deformed plane (a sphere).



Back in edit mode Tab, rotate the plane 90 degrees in the Z axis RZ90

Then rotate 90 degrees in X RX90


Move the plane up on the Y axis (this distance will be the radius of the sphere)


Press Space Bar type warp and select it.
On the warp settings change the WarpAngle to 180 degrees


Rotate the plane 90 degrees in the Y axis RY90 and warp again. This time make change the WarpAngle to 360 degrees.



At this point you should have a SPHERE!!


When you switch back to Object Mode your sphere will go back to a flat plane, don't worry you can control (and animate) the transformation by moving the value slider on the Shape Keys.




Then you can use UVs as the texture's coordinates in a material.